CC BY
13
Y^K 625.76.08
Aleksandr Zavyalov
Omsk State Technical University, Omsk, Russia
THE DYNAMICS OF THE INTERACTION PROCESSES BETWEEN WORKING ORGANS OF ROAD CONSTRUCTION MACHINES AND SOIL
The problem of mathematical modelling of the large-scale circulation of the Baltic Sea is considered. Marine hydrodynamics equations are written in the spherical coordinate system with a displaced point of the North Pole. The geographical North Pole is shifted to the vicinity of St. Petersburg to increase the spatial resolution of the Gulf of Finland. The free surface, sigma-coordinate primitive equation model under the Boussinesq, continuity, and hydrostatic assumptions is solved numerically. The problem of estimation of the pollution of some 'protected' marine sub-area by a passive tracer by means of the introducing an adjoint equation for the sensitivity function is formulated. The sensitivity function specifies the contribution of each basin point to the total pollution of the 'protected area'.
Keyword : Mathematical model; Baltic Sea circulation; splitting method; adjoint equation; sensitivity function; eco-socio-technical system
Introduction
The article presents decomposition, basic mathematical models of the interaction process between driven elements of road-making machines with the coupling medium which were developed by the author. These basic models provide working out of mathematical models of various complex processes of interactions under consideration. Examples of mathematical models could be found in the monograph [1].
Suppose mathematical models development satisfies the following assumption: coupling medium is plastically compressible. Based on such types of coupling medium movement as flat and cylindrical, developed is hierarchy system of mathematical models describing dynamics of the interaction under study. In contrast to dominating theories based on the statistics of loose grounds,
Динамика систем, механизмов и машин, № 1, 2014
which are seen as non-compressible, this approach is a relevant one as it lets model processes under study more precisely.
Consider the case when the device developed on the basis of an accepted soil model with mathematical description of some operations of the working process of road construction machines is applied.
Penetration
Let a certain working organ penetrate into the soil in the direction of OZ-axis, the origin of which is at the top of a penetrating body. We investigate two cases: when soil particles in front of the surface of a working organ produce a plane motion; the soil produce a cylindrical motion. The first case corresponds to the penetration of a wedge-like working organ into the soil. Let L = f (z) be the equation of a forming surface (fig. 1).
Fig. 1. A penetration diagram; Z - the direction of penetration; X - the direction of soil particles motion
The second case takes place at the penetration of a body with a conical or cylindrical surface, i.e. having a cylindrical symmetry. In this case R = f (z) is the equation of a generatrix of a penetrating body (see fig.2).
Case I. Let us assume that by virtue of unelastic character of the interaction with a body, soil particles will move along the planes perpendicular to OZ -axis (according to the analogy with the "hypothesis of plane cross-sections". The equations of motion and medium continuity according to will take on the form:
d 2 u DP
7 .
dt 2 dx du Го
1 +
(1) (2)
dx г
Ш t-'l
'Ж J
X
Fig. 2. A calculated diagram: I-I section; 2 - a soil motion region
Integrating the equation (2) of continuity for a variable X, we receive
x + u = \(x) + (p{t),
(3)
where y/(x) = ( dx
o K*)
Determined an arbitrar^' function of time such as on an timer boundary of a soil motion region. At x = 0
<p(t) = u(pj)=L(i)-rhus, x + u — y (x) +L(t) f
(5)
for a coordinate of an outer boundary we get correspondingly
x* = <p(x*)-L(t). (6)
x*
where
vCO
r
o 7 (*)
-dr.
Velocity mid acceleration for moving particles are obtained by differentiating the expression (5) with respect to time:
du _ ^ d~n _ £
a
St1
(7)
we get
Having integrated the equation of motion (1) for a variable x, subject to the expressions (7),
P = -yLx+P(p,f),
(s)
where PÍO, t) is the pressure of soil on the surface of penetrating body.
For an outer boundary of considered region (fig.2.2) tliis expression will take on the form
P* = J- L x I P(Oj), (9)
here P is the pressure of a soil on the boundary r . Form the expression (8) and (9) we get
P-PT --/0-L(x-x) (10)
Using the law of mass preservation and the theorem on the momentum on an outer boundary
/0z> = H¿>-¿), (11)
y^Dl* = P> (12)
where D is the velocity of an outer boundary; L - the velocity of particles oil an outer boundary; Po - pressure outside a motion region
From the equations (11), (12) following ratios are derived:
/o" - - a
P' =
1 -b(x*)
+ P0-,D =
1 -b(x*)
,, ro ,dyf 1
b(x ) = -— = (----)x = x
y(x ) dx x
(13)
Differentiating (5) with respect to time and putting x = x* ,u = 0, we get the expression for velocity' of particles on an external boundary of a motion region
T
it = L
Substituting the value P from (15), subject to (14), into (10), we receive
P = c
L2
1 -¿to
—-I(.v-.r )) + P0
(15)
Assuming that x = 0 in (15), we receive the expression of a soil pressure value oil the surface of a penetrating body
P~roil-b(x) For a case meeting the assumption [1,4,5]. when
r0
+Lx*) + P0_
(16)
Y(x)
= b = bl,
where b\ = const, the expression of pressure value P (15) is easily expressed as x - coordinate. Really, from (6) we get
jv = i^.v + L , hence,
1 -b,
Having substituted in (15), we shall liave
p=r0(-
L2
-L'x
LL
) + ?o
(17)
(18)
1 -b{ 1-Ä,
Oil the surface of a penetrating body (x = 0) the value of a soil pressure will be expressed as
7o fj-1
P =
1 -b,
a +ll)+Pü
(19)
hi the general case, when y = y(x) , density depends on x-coordinate; the relationship
between x* and x coordinate is set by the equation (6), then an explicit expression P is obtained through x-coordinate by the formula (15).
We investigate the motion of soil in an arbitrary section I-I (fi.g.3), which is off the soil surface, where a working organ is penetrating in the direction of 02 - axis by a depth H\(t\). Suppose, that at a point 0 of the tangency of the top of a body with the section at an mstant of time t\ , there appears a region or an excited motion in soil, and ait an instant t> t\ this region will be limited by values L and X . hi addition, the top of a body will be at the point 0>, which is off by the value Ii> if!.
Then from the moment the value L at the depth (H\ i is determined by a formula
L = f{z) = f[H{ f)-iiiOi)Lf>ii,
(20)
For the velocity and acceleration of soil particles attached to the surface of a body in a given section, we get
J.-J'H{H H,)H,L fH{H HJH2 + fH (H - Hx )H. (21)
Having replaced L and L ill the equations (2.15) and (2.16) by their values according to the formulae (21), we get the expressions of corresponding pressures in a section
p=nfH2{-)H- +f^(z)H)(x-x*) + P0, (22)
1 - ¿>(.v)
where Z = H - H\. Assuming that x = 0 in the equation (2.22)
P = +(/hW2 + /¿OO^K) + Po ■ (23)
An elementary site of the surface of a penetrating body with respect to its width B is expressed by a formula
ds = B^-f'2(z)dl. (24)
A force of resistance to the penetration, acting on an element of the surface in the direction of OZ - axis, can be written in the form
dW = P{sia(ar"crgf'(r)) + /¿o co^arcT^'(z)yb . (25)
Integrating the expression (2.25) with respect to the surface of a penetrated - into - the -soil body part, we get the value of a full force
H
W = j/>(sin(argi£f(-)) + ^COS(£J7Ctgf'(z))ds , (26)
0
here arctg f'(z) is the angle between a tangent and generatrix of a penetrating body and OZ -
Case 2. Let R =f(z) be an equation of a generatrix of a penetrating body (see fig.3), z - axis has the origin at the top and is directed along the axis of symmetry upward. From the moment ri of passing the top of a body in a considered section H\(t\) , a radius of a cross section of a body al this depth is determined by an expression
% = /oo fvm - Hi m ,r>rt. (27)
We assume, as in the first case, that at the point 0 of the tangency' of the top of a body' with a section al an instant of time fi , there appeal's the region of an excited motion in soil, and at the
moment / > ^ this region is limited by circles with radii R and r (see fig.3). Hie analysis of the motion of a pointed region is given in [1].
For the velocity* and acceleration of soil particles attached to the surface of a body in this section, we get formulae:
A - fk{H //,);/,, Ti, &(H-H,)H- + fB(H-HX)H. (28)
i'ftt I
Fig. 2.3. A calculated diagram We shall get the expressions of corresponding pressures acting on the surface of a penetrated body in the section Il\.
Ail elementary area of a body surface m the section will be expressed in tins case by a formula
ds = In ■ /Cz)V|ll-/'2(z>fc, (29)
and a fiill resistance to penetration, acting in the direction of a symmetry axis is determined by
W = 2jtJ uPf{ sy[\ +f'2 (z)d:, (30)
o
where u = sin (arctgf(z) - fi0 cos (arctgf (z)). The process of soil cutting
A developed in [1] mathematical body, describing a one-dimensional motion, allows to model the process of soil cutting.
Let us examine some cases of a mentioned process, in particular, a free cutting of wet soils and soils with normal moisture content.
In a free soil cutting the resistance of soil effects only a frontal face of a cutting organ. A cutting process stage, when a cutting organ is being penetrated into the soil by a certain resultant value h and a translational motion begins in the soil (fig .4), is taken as initial. Let fly) be an equation of a frontal surface of a cutting organ, a translational velocity of which is equal to Fi, and directed horizontally. Then the velocity of soil motion in an arbitrary section I-I will be quantitatively equal to:
V=Ve s*i(arctgfXy)) (31)
and directed along the normal to a considered surface.
ttttfyf'ht) Fig. 4. Calculation diagrams
The dynamics of soil, bemg in front of an elementary surface ds, is described by the equation [2]
fiv StA CP ct ox ex.
C utting af wet soils
A medium modelling such soils can be considered as uticompressible due to a large content of water in the volume of soil pores. Dynamic properties of soils with a factor of water saturation more than 0,5 lit the model adéquat elv [3].
74
Since the medium is uncompressiblef the velocity of soil particles will be a function of time t only; it means that
CLl
= 0_
(32)
CA
Let Ht) be the value of a displacement of a cutting organ m soil in the direction normal to its frontal surface. Then we can write
; OU v v(t )=L and — = L . ci
With respect to the expressions (32) and (33) will take on the form
CP
dx
Having integrated this equation for a r-vnriable, we get
P=-]lx + C,
(33)
(34)
(35)
where C is the constant of integration.
Since the soil in the region of an excited motion is not only uncompressible, but heterogeneous, i.e. y = y(x) , the density* being a function of x-coordinate, we write the expression of a soil pressure value on an external boundary of the region x as
P* = —y L x* + C. (36)
Basing oil the mam laws of mechanics, written for the parameters of the motion on the boundary of the region, it comes out that
r2 T
P* =
7
x =
1-b 1-b
On the frontal surface of a moving cuttmg organ x = L the formula (35) gives
P = -/LL + C.
(37)
(38)
Having excluded the constant of integration from the equations (36) and (38), and making use of ratios (37), we obtain the expression of a pressure value on a frontal surface
P=7o
L2 +LL l-b(L)
(39)
Hie formula (39) is an expression of a pressure value of soil on a frontal surface of a cuttmg organ in a considered arbitrary section I-I (see fig. 4). Then, if yo — yn(y) and P = P(y) are known functions, then horizontal and vertical components of the resistance to cutting, Fr and Fs correspondingly, are determined as
to cosecif
Fr= J P(y) sm(arcigf'<j))ds r (40)
o
75
FB = |>(.T) cos (arctgf(y))ds,
(41)
J /o(y) cos(arctgf'(yyJl+f2 (y)dy.
1 -ML) 1
(43)
It should be noted that the value I is limited by a certain value
— shear
(44)
Here Lihaar is such L - value, at which P leaches its maximum value, and shear and. shift of a soil mass occur. The formula (39) describes, in fact, the change in soil pressure value on a frontal surface for an arbitrary fragment of a cutting process defined by two successful shears.
Cutting of sails with normal moisture content
The soils, moisture content of which does not exceed 15%, otherwise, with a water saturation factor of no more than 0.5, are referred to this soil type. The soils with moisture content of 15% are referred to the most probable average soil type, more often developed with excavating machines.
Completing it with the equation of strain, which 111 the general case has the form
we get a closed system, the solutions of which can be presented in quadratures or solved numerically dependent on different conditions.
Having estimated the value of a frontal soil pressure, we can make use of the formulae (42) and (43) for detennniing horizontal and vertical components of resistance to cutting.
Note, if a cutting organ is curved, then the formulae (40) and (41) are transformed into following expressions
P = V2f{xJ),
(45)
D
(46)
D
(47)
where
cutting organ; r - an axis directed
peipendicular to a cross-section (see fi.g.4) of a cutting organ.
76
77
<p(t) = - sill 2psin ¡3{H - r0)dH /dt. (52)
TTie equation of motion in a chosen coordinate system will take on the from [1]
cGy! cH() - A31(cr1 - cj2 )cA3 / cff0 = yAydv / dt, (53)
where a and 03 are stresses acting along the lines Ho and 2. Because of the lack of motion along the line , the stress rg will be equal to:
1 ,
iJ2 =-(oi+cr3).
(54)
we get
where
The expression of a full acceleration dV/dt is presented in the form:
dvldt = duidt+K?udvldH0. (55)
Substituting here the expressions of a mass velocity of soil particles from the formula (50),
do fdt = A I dt- A3"A2<p2 (t),
dq>!dt=-[{H- rQ)d~H! dt7 + (dH i dt)2] sin 2/? sin p.
Taking into account, that the condition can be written in the form: = tJid - Lt) /(1 + U) + r0 /(1 + Li) , we get a following equation of motion (53):
cax 13H0 - OV/(H0 -r0) =
-[)A^/di + r0/(l + M)V(H0 -r0)-y<p2(t)/[AjA2-?-0)\
where
v = 2fifQ+fi). The solution of a given equation will be a function
(56)
(57)
(58)
(59)
(60)
tr, =e
- „"907) |
" J 0(7>q™dTj ], lb
where
(61)
17(77) = J(v/ Tf)dT}; TJ=H0 - r0l Tfo = Hb — r0, (62)
m,
where tjh is the value of a boundary region of soil particles motion.
Q07) = [A~l]d<p/dt + 7b /(1 + ,//)]//7 - y<p\t)!(\]AW)- №)
78
We shall receive the value of a mass particles velocity oil the boundary of a considered motion region substituting (62) into (51)
Uf, = (rfu'CIH i ryli) sin /3,
(64)
where r}w = H — r0 is the value of the coordinate of the surface of a penetrating body wedge.
The stress on the surface of the boundary of a soil motion region is determined by the relationship [6]
(65)
here JJb is the velocity of the boundary of a motion region, determined by the equation:
JJb = (dHb / dt) sin p , (66)
Pa - the stress in front of a motion region boundary, directed against Hg - axis (64) - (66).
Substituting the ratios (64) - (66) into the expression (63) and taking into account that on the surface of a wedge cfj = -P, then after integrating with respect to a variable rj with subsequent transformations, we get
P-Pa =Yq(d2Hfdt2)rjwsin2 mbfT}wy -l]/iA + + y0(dHfdf)2 sin2 jBibiv,-2)(dHb /dH)(jjb itjwY~1 -- [(nb /n„Y~2 - 1] + (V - 2) /Vt076 /Tjwy - 1 ]}/[b + (v - 2)] -+ -l\[Pa+r0/v(l + /i)].
(67)
For calculating pressure, acting on the surface of a penetrating wedge, it is necessary to find a motion region boundary, tji - wedge surface. coordinates relationship. This relationship follows from the law of conserving a medium mass enclosed mto an elementary volume, and has the form [2]
% = 'AW'1 + b =w
Recording the dependence (68) m a detailed fomi
1/2
(68)
(69)
M^b'
differentiating (69) with respect to time
dHb / dt = (dH i dt)Q.-b)1'2 = a112 dH / dt (70)
and substituting the expressions (69), (70) mto the formula (67), we get
P - Pa = i,w (d2H; dt2)r0 (a"/2 -1) sill2 plib) + (dH ! dtfy^v - 2) x x (a1'12 - \)!v+b{v - 2)if'2 - (a"21 - l)]sin2 (5 ¡[b, (v - 2)] + [avn -1) x (7i)
Havmg recorded the condition [1] in the form
-Olgp + C
(72)
and. taking into account that, Pa— <7, and r0 = 2c cosp we get
Pa ={2cos/?-l)-c-ctgp (73)
According to an adopted model of filling a scraper's bucket with soil, we define the value of resistance to filling,. WH, as an integral
WH={\ + fictgP)\\{P-Pa)$mpch, (74)
j
where ds = Bdrjv cosfi - an element of the surface of a penetrating wedge; B - the width of a bucket.
Since a considered surface consists of two parts then, substituting a doubled ds into (74) 3 we
H
WH = i? sin 2/7(1 + iieigj3)\{P - Pa)d qb. (75)
o
Or with respect to (71), putting H = —ctgp , we get the expression of the resistance to fillmg at a given thickness of cuttings, h , and a bucket width B
WH = 5(1 + fJctgP) [A + k(dH! dtf + RH{d2H i dt2 )]H, (76)
where A = 2(a1:2 - l)(cosecp - ctgp)c cospsmi/?; (77)
k = y0[(i ■- ■- 2)(av'! 2 -1)/1'■+ ^(v- l)d '2 - (ao/:2 - l)]sin2 psin 2p![b{v - 2)]. (78)
R = y0(av/2 -l)siii2 p%in2/i/(2t-S1); (79)
H = -hctgp, (SO)
2
b-FlH-J - the dependence is set experimentally.
Hie expressions (77), (78) and (79) present a static, kinematic and dynamic components of a resistance-to-filliiig value, fYh, correspondingly'.
In a considered model of a filling process the forces of a lateral widening are being ignored, however., according to data [7]? they make up 5-7% of a full resistance to filling A doubled angle over the interval [15° ; 25°], /?, is associated in a physical sense, with a shearing angle for soil cuttings, dependent on their thickness and physico-mechanical properties of soil, the angle 30c<
1
F< 50Cl correspondmgly. Putting ¡3 = — y/.. d H ' dt = const, we get
2
for low velocities 0.1,..., 1.0 (m;s) and not thick cuttings h ~ 0.1 m.; WH values close to the results being obtained, according to the method [7].
Some results of formulae analysis (76) - (80) are presented in fig.6. The calculations were made with following data:
e=30 JtRn; B=2.52m\ pa=26°; p=31°
;14 t/rn 5=0.85; Hg=l6&m; h=0.\m
SO
(5 17S so 22.S fi''/tf.
Fig.G. Dependences of the resistance of filling oil a penetrating wedge angle value and rtie rate of digging: 1; % 3; 4: 5 - at dH/dt = 0.5: 1.0; 1.5; 2.0 and 2.5 m/.s correspondingly; 6 - by [7]
A given model of the process of filling a scraper's bucket with soil allows to define the following:
1. Hie resistance to filling at any instant of time of a digging process provided that the relations Hn = Ha (t) and b = F(Hu) are set.
2. The effect of kinematic and dynamic factors on a filling process and the process of soil digging with a scraper as a whole.
3. The dependence of the resistance to filling on physico-mechanical properties of soil and digging process conditions
It should be noted that a mathematical model of the process of filling a bucket is obtained on condition that mechanical properties of soil inside a soil motion region (fig.5) are expressed as ratios.
2.4. Fanning and displacement of a soil prism
For determining a resistance-to-displacement value of a soil prism we make use of the relationship. Then a value of a normal pressure on an element of a surface of a working organ (fig.7) can be presented by the expression
P=(p*?-f{x\f), (SI)
where P is a normal pressure value, u - the velocity of soil particles motion, fix*. t) - a certain
differentiatiable function, x - a coordinate, t - time.
Hie expression (81) is a cooperation of two equations, describing a dynamic state of soil -
plasticity and strains. If a velocity value v = ¿j(.y* . f) can be presented as
u\x*,0 = ul(x*yu2(t), (82)
here Uj (jc ), u2 (f) - certain given functions, i.e. variables can be divided, then a general solution of a partial differential equation interpreting a one-dimensional motion of soil in a considered local
, * „So* „ cP
r(x,0(— + v —) = -—, (S3)
ct av cv
has the form
y* * I-~—
where W = \ P(x'. F)[siii(argtgx'y ■+ fgp0 cos(arctgx'¥y\yj 1 - (.Vy) flY.
F*=(rs-T) - a height of a soil prism according to a chosen coordinate system, where a top point of a working organ (see fig. 7) is taken as the origin of the coordinates, defining its height, s;
po~ the angle of an external friction; ^jl + {Xy^ dY— a profile differential v =x(y), m - soil mass in a prism; ud and if - velocities for displacement of a working organ and the particles of an attached soul mass correspondingly. A vertical component of a considered resistance will be determined somewhat simpler
W* = J P(.v*. Y)[cos(mctgx'r) + fgp0 sm(arctgx^ljl + (x'r)2dx
(89)
Determine the expression of a soil prian mass proceeding from following considerations
T*
\ydV, (90)
1)1
where V- the volume of soil in a prism, the expression of which (fig .7) will have the form
1
Then
?
dV = BU?¥-Y*dY\
(91)
(92)
provided that the pressure of soil ni a prism and the density aie distributed according to a certain linear law suppose
where / - a coefficient of linearity, and
f(x\Y\t) = ky,
(93)
(94)
here k - a constant dependent on physico-mechamcal properties of soil.
Hie expression of a pressure value on the boundary x = x.(y), subject to the formula (SI), (93), (94), takes the form
P = kyo
where u = u (0, f) - a modular machine speed.
P = k-x-YV.
(95)
(96)
Returning to the formula (90) and substitutmg the expressions (92) and (93), we receive
Y*
111 = X-B- tg^V [ Y'fi - Y VA' . (97)
ïi
83
On integrating, there will be
m = I В -- 27* )Y. (98)
6
Tlie process of prism forming depends on a distance and digging speed, hi this connection we introduce a coefficient с. setting a relationship between a modular speed of a mad-construction machine and a horizontal rate of inclement in the base of a soil prism, if we consider that S —
const, then the change in a prism height.. Y" . dependent on time, t, is expressed as
Y * = (99)
n
where t\ - an instant or time corresponding to the beginning of the process ox prism forming.
dm
With respect to the latter formula, the rate of chancing a soil mass in a prism, - is
dt
registered as
- = H-B-£-v(s-Y*)-7*. (100)
dt
Thus, the formulae (2.87) - (2.100) present an algontlmi, describing the dynamics of a prism forming process and allowing to determine the value of resistance to displacement of a soil prism by a working organ of a road construction machine having an arbitrary configuration x = x(y), at any instant of time.
Tlie analysis of this value was done by an example of a scraper with a geometrical capacity of 8 m\ Under a soil prism we understand a traction prism in front of a scraper's bucket shutter. The algoritlim is realized under following parameters quantities:
s=1.25m; ;=25xl(T3; pc=0.4; 3=2.43;
W=0.61; t, =8s; k=l; /=4,5.
The quantity of x-coefficient is received fiom initial conditions:
P=5xl0~i mPa; Y=1.25m; v=lm/s; k=l.
The results of the analysis are shown in figures 8 and 9.
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